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1. Dirac geometry II: coherent cohomology

   https://doi.org/10.1017/fms.2024.2  

   Hesselholt, Lars; Pstrągowski, Piotr (January 2024, Forum of Mathematics, Sigma)

   Abstract Dirac rings are commutative algebras in the symmetric monoidal category of$$\mathbb {Z}$$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger$$\infty $$-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to$$\operatorname {MU}$$and$$\mathbb {F}_p$$in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves. 

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2. Simple closed curves in stable covers of surfaces

   https://doi.org/10.1090/tran/8940  

   Salter, Nick (May 2023, Transactions of the American Mathematical Society)

   Let $f:X\to <\#comment/>Y$be a regular covering of a surface $Y$of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on $Y$, possibly after passing to a “stabilization”, i.e. a $G$-equivariant embedding of covering spaces $X\hookrightarrow <\#comment/>X^{+}$. 

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3. Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

   https://doi.org/10.4171/JEMS/1407  

   An, Jing; Henderson, Christopher; Ryzhik, Lenya (April 2025, Journal of the European Mathematical Society)

   We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength\beta\in\mathbb{R}. This equation exhibits a transition from pulled to pushed front behavior at\beta_{c}=2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a positionm_{\beta}(t)and study the asymptotics of the front locationm_{\beta}(t). When\beta < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson:m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1)ast\to\infty. This form is typical of pulled fronts. When\beta > 2, the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1)with c_{*}(\beta)=\beta/2+2/\beta, which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2, the expansion changes tom_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1), reflecting the “pushmi-pullyu” nature of the front. The arguments for\beta<2rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case \beta>2relies on standard pushed front techniques. The proof in the case\beta=\beta_{c}is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at\beta_{c}=2and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights. 

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4. Dimension-free discretizations of the uniform norm by small product sets

   https://doi.org/10.1007/s00222-024-01306-9  

   Becker, Lars; Klein, Ohad; Slote, Joseph; Volberg, Alexander; Zhang, Haonan (December 2024, Inventiones mathematicae)

   Abstract Let$$f$$ $f$be an analytic polynomial of degree at most$$K-1$$ $K-1$. A classical inequality of Bernstein compares the supremum norm of$$f$$ $f$over the unit circle to its supremum norm over the sampling set of the$$K$$ $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for$$L^{p},1\le p\leq \infty $$ $L^p,1\le p\le \infty$norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of$$\deg (f)$$ $deg\left(f\right)$and is instead governed by the maximumindividualdegree of$$f$$ $f$;i.e., the largest degree of$$f$$ $f$when viewed as a univariate polynomial in any coordinate. For example, we find that for$$n$$ $n$-variate analytic polynomials$$f$$ $f$of degree at most$$d$$ $d$and individual degree at most$$K-1$$ $K-1$,$$\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$$ ${\parallel f\parallel }_{L^{\infty }\left(D^n\right)}\le C{\left(X\right)}^d{\parallel f\parallel }_{L^{\infty }\left(X^n\right)}$for any fixed$$X$$ $X$in the unit disc$$\mathbf{D}$$ $D$with$$|X|=K$$ $\left|X\right|=K$. The dependence on$$d$$ $d$in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant$$\mathcal{O}(\log K)^{2d}$$ $O{(logK)}^{2d}$. 

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5. Integral models for spaces via the higher Frobenius

   https://doi.org/10.1090/jams/998  

   Yuan, Allen (January 2023, Journal of the American Mathematical Society)

   We give a fully faithful integral model for simply connected finite complexes in terms of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$is the data of its Spanier-Whitehead dual, as an ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\mathrm{\infty <\#comment/>}$-category of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we callpartial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of ${\mathbb{F}}_p$up to $p$-completion. 

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